In this article, we focus on the existence and topology of edge states in
AB-stacked bilayer graphene (BLG) ribbon [1]}, [2]} by connecting them
with \(k_y\) -parametrized SSH ladder. Earlier works are mainly concentrated on
the existence of edge states of BLG under specific edge conditions [3]}, [4]}, or the behavior of edge states when various symmetry-breaking
terms are added [5]}, [6]}. Here the AB-stacked BLG ribbon we discuss
involves three conventional edges (zigzag, bearded, and armchair, as shown
in Fig. 1), whose bulk 2D system always has zero Chern number. Interlayer
next nearest neighbor (NNN) interaction and interlayer bias are considered
in terms of their influence on the topology of SSH ladder. A detailed
topological classification based on discrete symmetry [7]} and
topological invariants calculation for effective 1D bulk Hamiltonian of SSH
ladder \(H\left( k_{y},k\right) \)  parameterized by \(k_{y}\)  of AB-stacked BLG
ribbon with various types of edge are performed, as shown in Table REF . It shows the zero-energy edge states can only exist when chiral
symmetry is preserved for \(H\left( k_{y},k\right) \)  and can appear in the
whole \(k_{y}\)  region when two layers of BLG ribbon have zigzag and bearded
edge, respectively. On the other hand, straightforward calculation shows
that unexpected edge states can exist in the gap within bulk bands that are
away from the Fermi level. These edge states are unprotected by the chiral
symmetry and are dependent on the specific edge configurations of BLG
ribbon. Interlayer bias is included in our discussion as it explicitly
breaks the chiral symmetry responsible for the existence of zero-energy edge
states. However, edge states still exist after this chiral symmetry breaking
as non-zero energy states.
